More later, but just a brief remark – I think that one issue is that the top ~200 mathematicians are of such high intellectual caliber that they’ve plucked all of the low hanging fruit and that as a result mathematicians outside of that group have a really hard time doing research that’s both interesting and original.
I’m not sure what gives you this impression. In my own field (number theory) I don’t get that feeling at all. There may not be much low hanging fruit, but there’s more than enough for people who aren’t that top 200 to do very useful work.
I’d certainly defer to you in relation to subject matter knowledge (my knowledge of number theory really only extends through 1965 or so), but this is not the sense that I’ve gotten from speaking with the best number theorists.
When I met Shimura, he was extremely dismissive of contemporary number theory research, to a degree that seemed absurd to me (e.g. he characterized papers in the Annals of Mathematics as “very mediocre.”) I would ordinarily be hesitant to write about a private conversation publicly, but he freely and eagerly expresses his views freely to everyone who he meets. Have you read The Map of My Life? He’s very harsh and cranky and perhaps even paranoid, but that doesn’t undercut his track record of being an extremely fertile mathematician. I reflected on his comments and learned more over the years (after meeting with him in 2008) his position came to seem progressively more sound (to my great surprise!).
A careful reading of Langlands’ Reflexions on receiving the Shaw Prize hints that he thinks that the methods that Taylor and collaborators have been using to prove theorems such as the Sato-Tate conjecture won’t have lasting value, though he’s very guarded in how he expresses himself. I remember coming across a more recent essay where he was more explicit and forceful, but I forget where it is (somewhere on his website, sorry, I realize that this isn’t so useful). It’s not clear to me that Taylor would disagree – he may explicitly be more committed to solving problems in the near term than by creating work of lasting value.
One can speculate these views are driven by arrogance, but they’re not even that exotic outside of the set of people who have unambiguously done great work. For example, the author of the Galois Representations blog, who you probably know of, wrote in response to Jordan Ellenberg:
That said, there is a secondary argument which portrays mathematics as a grand collective endeavour to which we can all contribute. I think that this is a little unrealistic. In my perspective, the actual number of people who are advancing mathematics in any genuine sense is very low. This is not to say that there aren’t quite a number of people doing interesting mathematics. But it’s not so clear the extent to which the discovery of conceptual breakthroughs is contingent on others first making incremental progress. This may sound like a depressing view of mathematics, but I don’t find it so. Merely to be an observer in the progress of number theory is enough for me — I know how to prove Fermat’s Last Theorem, how exciting is that?
apparently implicitly characterizing his own work as insignificant. And there aren’t very many number theorists as capable as him.
Replying separately so it isn’t missed. I wonder also how much of these issues is the two cultures problem that Gowers talks about. The top people conception seems to at least lean heavily into the theory-builder side.
I agree with you, and there are strong problem solver types who conceptualize mathematical value in a different way from the people who I’ve quoted and from myself.
Still, there are some situations where one has apples-to-apples comparisons.
There’s a large body of work giving unconditional proofs of theorems that would follow from the Riemann hypothesis and its generalizations. Many problem solvers would agree that a proof of the Riemann hypothesis and its generalizations would be more valuable than all of this work combined.
We don’t yet know how or when the Riemann hypothesis will be proved. But suppose that it turns out that Alain’s Connes’ approach using noncommutative geometry (which seems most promising right now, though I don’t know how promising) turns out to be possible to implement over the next 40 years or so. In this hypothetical. What attitude do you think that problem solvers would take to the prior unconditional proofs of consequences of RH?
I agree with you, and there are strong problem solver types who conceptualize mathematical value in a different way from the people who I’ve quoted and from myself.
Yes, but at the same time (against my earlier point) the best problem solvers are finding novel techniques that can be then applied to a variety of different problems- that’s essentially what Gowers seems to be focusing on.
There’s a large body of work giving unconditional proofs of theorems that would follow from the Riemann hypothesis and its generalizations. Many problem solvers would agree that a proof of the Riemann hypothesis and its generalizations would be more valuable than all of this work combined.
I’m not sure I’d agree with that, but I feel like I’m in the middle of the two camps so maybe I’m not relevant? All those other results tell us what to believe well before we actually have a proof of RH. So that’s at least got to count for something. It may be true that a proof of GRH would be that much more useful, but GRH is a much broader idea. Note also that part of the point of proving things under RH is to then try and prove the same statements with weaker or no assumptions, and that’s a successful process.
What attitude do you think that problem solvers would take to the prior unconditional proofs of consequences of RH?
I’m not sure. Can you expand on what you think would be their attitude?
I’d certainly defer to you in relation to subject matter knowledge (my knowledge of number theory really only extends through 1965 or so), but this is not the sense that I’ve gotten from speaking with the best number theorists.
I’ll volunteer another reason not to necessarily pay attention to my viewpoint: I’m pretty clearly one of those weaker mathematicians, so I have obvious motivations for seeing all of that side work as relevant.
I suspect that one can get similar viewpoints from people who more or less think the opposite but that they aren’t very vocal because it is closer to being a default viewpoint, but my evidence for this is very weak. It is also worth noting that when one does read papers by the top named people, they often cite papers from people who clearly aren’t in that top, using little constructions or generalizing bits or the like.
I’ll volunteer another reason not to necessarily pay attention to my viewpoint: I’m pretty clearly one of those weaker mathematicians, so I have obvious motivations for seeing all of that side work as relevant.
I’ll note that I think that there are people other than top researchers who have contributed enormously to the mathematical community through things other than research. For example, John Baez is listed amongst the mathematicians who influenced MathOverflow participants the most, in the same range as Fields medalists and historical greats, based on his expository contributions.
It is also worth noting that when one does read papers by the top named people, they often cite papers from people who clearly aren’t in that top, using little constructions or generalizing bits or the like.
Yes, this is true and a good point. It can serve as a starting point for estimating effect sizes.
I’m not sure what gives you this impression. In my own field (number theory) I don’t get that feeling at all. There may not be much low hanging fruit, but there’s more than enough for people who aren’t that top 200 to do very useful work.
I’d certainly defer to you in relation to subject matter knowledge (my knowledge of number theory really only extends through 1965 or so), but this is not the sense that I’ve gotten from speaking with the best number theorists.
When I met Shimura, he was extremely dismissive of contemporary number theory research, to a degree that seemed absurd to me (e.g. he characterized papers in the Annals of Mathematics as “very mediocre.”) I would ordinarily be hesitant to write about a private conversation publicly, but he freely and eagerly expresses his views freely to everyone who he meets. Have you read The Map of My Life? He’s very harsh and cranky and perhaps even paranoid, but that doesn’t undercut his track record of being an extremely fertile mathematician. I reflected on his comments and learned more over the years (after meeting with him in 2008) his position came to seem progressively more sound (to my great surprise!).
A careful reading of Langlands’ Reflexions on receiving the Shaw Prize hints that he thinks that the methods that Taylor and collaborators have been using to prove theorems such as the Sato-Tate conjecture won’t have lasting value, though he’s very guarded in how he expresses himself. I remember coming across a more recent essay where he was more explicit and forceful, but I forget where it is (somewhere on his website, sorry, I realize that this isn’t so useful). It’s not clear to me that Taylor would disagree – he may explicitly be more committed to solving problems in the near term than by creating work of lasting value.
One can speculate these views are driven by arrogance, but they’re not even that exotic outside of the set of people who have unambiguously done great work. For example, the author of the Galois Representations blog, who you probably know of, wrote in response to Jordan Ellenberg:
apparently implicitly characterizing his own work as insignificant. And there aren’t very many number theorists as capable as him.
Replying separately so it isn’t missed. I wonder also how much of these issues is the two cultures problem that Gowers talks about. The top people conception seems to at least lean heavily into the theory-builder side.
I agree with you, and there are strong problem solver types who conceptualize mathematical value in a different way from the people who I’ve quoted and from myself.
Still, there are some situations where one has apples-to-apples comparisons.
There’s a large body of work giving unconditional proofs of theorems that would follow from the Riemann hypothesis and its generalizations. Many problem solvers would agree that a proof of the Riemann hypothesis and its generalizations would be more valuable than all of this work combined.
We don’t yet know how or when the Riemann hypothesis will be proved. But suppose that it turns out that Alain’s Connes’ approach using noncommutative geometry (which seems most promising right now, though I don’t know how promising) turns out to be possible to implement over the next 40 years or so. In this hypothetical. What attitude do you think that problem solvers would take to the prior unconditional proofs of consequences of RH?
Yes, but at the same time (against my earlier point) the best problem solvers are finding novel techniques that can be then applied to a variety of different problems- that’s essentially what Gowers seems to be focusing on.
I’m not sure I’d agree with that, but I feel like I’m in the middle of the two camps so maybe I’m not relevant? All those other results tell us what to believe well before we actually have a proof of RH. So that’s at least got to count for something. It may be true that a proof of GRH would be that much more useful, but GRH is a much broader idea. Note also that part of the point of proving things under RH is to then try and prove the same statements with weaker or no assumptions, and that’s a successful process.
I’m not sure. Can you expand on what you think would be their attitude?
I’ll volunteer another reason not to necessarily pay attention to my viewpoint: I’m pretty clearly one of those weaker mathematicians, so I have obvious motivations for seeing all of that side work as relevant.
I suspect that one can get similar viewpoints from people who more or less think the opposite but that they aren’t very vocal because it is closer to being a default viewpoint, but my evidence for this is very weak. It is also worth noting that when one does read papers by the top named people, they often cite papers from people who clearly aren’t in that top, using little constructions or generalizing bits or the like.
I’ll note that I think that there are people other than top researchers who have contributed enormously to the mathematical community through things other than research. For example, John Baez is listed amongst the mathematicians who influenced MathOverflow participants the most, in the same range as Fields medalists and historical greats, based on his expository contributions.
Yes, this is true and a good point. It can serve as a starting point for estimating effect sizes.